Mermin-Wagner Theorem and Lower Critical Dimension

Whether a many-body system can order at low temperature turns on two things: the spatial dimension $d$ , and whether the symmetry it would break is discrete or continuous. The Mermin–Wagner theorem gives a sharp, rigorous verdict for one crucial class: a continuous symmetry, with short-range interactions, cannot be spontaneously broken at any finite temperature in $d\le 2$ . The spontaneous magnetization of a two-dimensional Heisenberg ferromagnet, or the condensate amplitude of a 2D superfluid, is exactly zero — not small, but zero. The engine behind this is the Goldstone theorem. Breaking a continuous symmetry produces gapless modes — spin waves, the phase of a condensate — whose energy vanishes as $k\to 0$ . The thermal population of these soft modes carries an infrared weight $\langle\theta^2\rangle\propto\int\mathrm{d}^d k/k^2$ , and this integral diverges at small $k$ whenever $d\le 2$ . Unbounded angular fluctuations average the order parameter to zero. A discrete symmetry has no such soft mode: its cheapest excitation is a domain wall with a finite energy per unit area, so no divergence appears and order survives down to $d=1$ (where it holds only at $T=0$ ). The organizing concept is the lower critical dimension $d_\ell$ — the dimension at or below which thermal fluctuations destroy long-range order for all $T>0$ . For a discrete order parameter, $d_\ell=1$ ; for a continuous one, $d_\ell=2$ . Exactly at $d_\ell$ the behavior is marginal and rich: the 1D Ising model orders only at $T=0$ , and the 2D $XY$ model, though it cannot break its symmetry, exhibits the Kosterlitz–Thouless transition into a phase of quasi-long-range order. This article builds the spin-wave estimate that exposes the infrared divergence, states the rigorous Bogoliubov-inequality version, contrasts it with the domain-wall (Peierls) argument for discrete symmetry, and surveys the loopholes that restore order.

Key takeaways

The Mermin–Wagner theorem: in dimensions $d\le 2$ a continuous symmetry cannot be spontaneously broken at any $T>0$ with short-range interactions, because long-wavelength Goldstone fluctuations carry infinite phase-space weight.
The mechanism is the infrared divergence of $\int\mathrm{d}^d k/k^2$ : gapless Goldstone modes destroy any putative order whenever $d\le 2$ .
This distinguishes continuous models (Heisenberg, $XY$ , $O(n\ge2)$ ), which have $d_\ell=2$ , from discrete models (Ising), which order in $d=2$ as Peierls and Onsager proved.
The lower critical dimension $d_\ell$ is where order is destroyed: $d_\ell=1$ for discrete symmetry, $d_\ell=2$ for continuous symmetry.
A crucial loophole: the 2D $XY$ model still undergoes the Berezinskii–Kosterlitz–Thouless transition into a quasi-long-range-ordered phase, even with no true order.
Key idea: gapless Goldstone modes from a broken continuous symmetry have a thermal occupation that diverges as $\int\mathrm{d}^d k/k^2$ in the infrared; this destroys the order parameter for $d\le 2$ . Discrete symmetries have no such soft mode, so they order down to $d=1$ (at $T=0$ ).

The core insight

Gapless Goldstone modes from a broken continuous symmetry have a thermal occupation that diverges as $\int\mathrm{d}^d k/k^2$ in the infrared; this divergence destroys the order parameter for $d\le 2$ . Discrete symmetries have no such soft mode, so they order down to $d=1$ (at $T=0$ ).

Where this lives today

The theorem is a working design rule for two-dimensional materials. Isotropic Heisenberg monolayers cannot order at $T>0$ , so every real 2D magnet that does order — monolayer CrI $_3$ , Fe $_3$ GeTe $_2$ — must owe its Curie temperature to magnetic anisotropy, which opens a gap $\Delta$ in the spin-wave spectrum, $\epsilon(k)\to\Delta+a^2k^2$ , cutting off the infrared divergence. The measured Curie temperatures track the anisotropy gap, a direct experimental fingerprint of the theorem's loophole, and the discovery of 2D ferromagnetism in 2017 was celebrated precisely because it navigates this constraint. The same logic reaches beyond magnetism. Phonons are the Goldstone modes of broken translational symmetry, so a strictly 2D crystal has no true long-range positional order (the Peierls–Landau instability), only quasi-order — and 2D melting proceeds through the KTHNY scenario of dislocation and disclination unbinding, seen in colloidal monolayers. Hohenberg's quantum cousin of the theorem forbids a true Bose–Einstein condensate in 2D at $T>0$ , yet 2D Bose gases are superfluid below $T_{\mathrm{BKT}}$ with the universal Nelson–Kosterlitz stiffness jump, now measured directly in ultracold-atom experiments. Identifying a model's symmetry (discrete versus continuous) and interaction range immediately fixes $d_\ell$ and tells an experimentalist whether a finite-temperature transition is even possible.

The physical picture

Picture a 2D ferromagnet that has chosen a direction. To rotate its spins gently over a region of size $L$ costs an energy $\sim J(\delta\theta/L)^2\times L^2 = J\,\delta\theta^2$ — independent of $L$ . Because arbitrarily large, arbitrarily soft twists cost essentially the same finite energy, thermal noise excites all of them, and the accumulated angular wandering grows without bound. The magnetization, an average of $\cos\theta$ over wildly fluctuating $\theta$ , averages to zero. Contrast a 2D Ising magnet. To reverse a region you must pay a domain-wall energy proportional to the wall's length, i.e. proportional to $L$ , not constant. Large flips are exponentially suppressed, so the magnetization survives. The entire difference between $d_\ell=2$ and $d_\ell=1$ is the difference between "cost independent of size" (a Goldstone twist) and "cost growing with size" (a domain wall).

Figure

Two ways to disorder a magnet: a soft twist (constant cost) vs a domain wall (cost grows with L)

Everything is contained in one integral: $\langle\theta^2\rangle\sim\frac{k_BT}{J}\int\frac{\mathrm{d}^d k}{k^2}$ . The measure $\mathrm{d}^d k=k^{d-1}\mathrm{d}k\times(\text{angles})$ turns this into $\int_0 k^{d-3}\mathrm{d}k$ , which converges at $k\to 0$ only for $d>2$ . The borderline $d=2$ gives a logarithm — the seed of the marginal Kosterlitz–Thouless physics. The punchline is the growth of the angular fluctuation with system size. In three dimensions $\langle\delta\theta^2\rangle$ saturates to a finite value as $L\to\infty$ — order survives, merely reduced by spin waves. In two dimensions it climbs without bound as $\ln L$ — slowly, but forever — and that logarithmic runaway is exactly what erases the order parameter. The 2D log-divergence, sitting between 3D saturation and the violent 1D linear growth, is the whole theorem in one curve.

Figure

Angular fluctuation ⟨δθ²⟩ vs system size L: 2D diverges as ln L, 3D saturates

The core equations

The $O(n)$ model generalizes Ising ( $n=1$ ) to $n$ -component classical unit spins. For $n\ge2$ the symmetry group $O(n)$ is continuous, and it is precisely the continuous case to which Mermin–Wagner applies.

H=-J\sum_{\langle ij\rangle}\mathbf{S}_i\cdot\mathbf{S}_j,\qquad |\mathbf{S}_i|=1,\ \mathbf{S}_i\in\mathbb{R}^{n} \qquad (1)

where: $J>0$ — ferromagnetic exchange; $\mathbf{S}_i$ — $n$ -component unit spin; $n$ — number of components. For $n=1$ the spin is $\pm1$ (discrete $\mathbb{Z}_2$ , Ising) with no continuous rotation and no Goldstone mode; for $n\ge2$ the ordered state selects a direction, leaving $n-1$ Goldstone spin-wave modes. Expanding about an ordered state in the transverse (Goldstone) fields $\boldsymbol{\pi}$ gives a free Gaussian theory whose energy depends only on gradients — the defining feature of a Goldstone field.

H\approx\text{const}+\frac{J}{2}\sum_{\langle ij\rangle}(\boldsymbol{\pi}_i-\boldsymbol{\pi}_j)^2 = \frac{J}{2}\sum_{\mathbf{k}}\epsilon(\mathbf{k})\,|\boldsymbol{\pi}_{\mathbf{k}}|^2,\qquad \epsilon(\mathbf{k})\xrightarrow{k\to0}a^2 k^2 \qquad (2)

where: $\boldsymbol{\pi}_i\in\mathbb{R}^{n-1}$ — transverse fluctuation fields; $\epsilon(\mathbf{k})=\sum_\delta(1-\cos\mathbf{k}\cdot\boldsymbol\delta)\to a^2k^2$ ; $a$ — lattice spacing. The $\epsilon\to a^2k^2$ softening at small $k$ is the gaplessness that Goldstone's theorem guarantees. Equipartition gives each mode a variance with a dangerous $1/k^2$ infrared enhancement. Summing over modes yields the local fluctuation, whose convergence decides ordering.

\langle\boldsymbol{\pi}_i^2\rangle=\frac{(n-1)k_BT}{J}\int\frac{\mathrm{d}^dk}{(2\pi)^d}\,\frac{1}{a^2k^2} \qquad (3)

where: $n-1$ — number of transverse components; the $1/k^2$ comes from $1/\epsilon(\mathbf{k})$ . Read physically: this integral is the total angular wandering; if it diverges, the direction the system "chose" is washed out. The infrared behavior of the radial integral is the whole verdict. Reducing $\mathrm{d}^dk=k^{d-1}\mathrm{d}k\times\text{angles}$ leaves $k^{d-3}\mathrm{d}k$ .

\int_{1/L} k^{d-1}\,\frac{\mathrm{d}k}{k^2}=\int_{1/L} k^{d-3}\,\mathrm{d}k \propto \begin{cases}\text{finite}&d>2\\ \ln L&d=2\\ L^{2-d}&d<2\end{cases} \qquad (4)

where: the lower cutoff $1/L\to0$ in the thermodynamic limit. For $d\le2$ the angular variance diverges; for $d>2$ it is bounded. This single line is the mathematical content of the theorem. The order parameter is suppressed by a Debye–Waller-like factor of the fluctuation, driving the magnetization to zero for $d\le2$ .

\langle\mathbf{S}_i\rangle=\hat{e}_1\,\langle\sqrt{1-\boldsymbol{\pi}^2}\rangle\approx\hat{e}_1\,e^{-\frac12\langle\boldsymbol{\pi}_i^2\rangle}\xrightarrow{d\le2}0 \qquad (5)

where: $e^{-\frac12\langle\pi^2\rangle}$ — Gaussian suppression from angular fluctuations. Unbounded $\langle\pi^2\rangle$ sends the magnetization exactly to zero — the harmonic estimate captures the rigorous result. The rigorous version replaces the harmonic estimate with the Bogoliubov inequality, valid for the full nonlinear model with no approximation.

m^2\le \frac{\text{const}}{\displaystyle\int \mathrm{d}^dk\,(h+k^2)^{-1}} \qquad (6)

where: $m$ — magnetization in the $h\to0$ limit; $h$ — symmetry-breaking field, sent to zero after the thermodynamic limit. The denominator diverges as $h\to0$ for $d\le2$ , forcing $m\to0$ — the exact analogue of the spin-wave integral, but now a theorem.

Why continuous and discrete symmetries differ

Derivation ASpin-wave destruction of continuous order in d ≤ 2

We want the harmonic (Gaussian) estimate that exhibits the infrared divergence destroying continuous order in two dimensions and below. The strategy: start from the $O(n)$ ferromagnet, expand around an ordered state to quadratic order in the transverse Goldstone fields, use equipartition to get each mode's variance, sum over the Brillouin zone, and analyze the small- $k$ (infrared) end of the resulting integral, which is where the fate of order is decided. The one thing to keep your eye on is the exponent $k^{d-3}$ — its integrability at $k=0$ is the entire question.

Step 1. Expand the exchange energy to quadratic order in the transverse fields; the result is a free-field Hamiltonian,

H\simeq\frac{J}{2}\sum_{\langle ij\rangle}(\boldsymbol{\pi}_i-\boldsymbol{\pi}_j)^2=\frac{J}{2}\sum_{\mathbf{k}}\epsilon(\mathbf{k})|\boldsymbol{\pi}_{\mathbf{k}}|^2,

with $\epsilon(\mathbf{k})\to a^2k^2$ at long wavelength. Why: the Goldstone fields enter only through differences (gradients), so a uniform rotation costs nothing and the dispersion vanishes as $k\to0$ — the defining softness.

Step 2. Equipartition assigns each of the $n-1$ transverse Gaussian modes the variance

\langle|\boldsymbol{\pi}_{\mathbf{k}}|^2\rangle=\frac{(n-1)k_BT}{J\epsilon(\mathbf{k})}.

Why: every quadratic mode carries $\tfrac12 k_BT$ ; the $1/\epsilon\sim1/k^2$ is the infrared amplification that makes long wavelengths dominate.

Step 3. The real-space variance is the integral of the mode variance over the Brillouin zone,

\langle\boldsymbol{\pi}_i^2\rangle=\frac{(n-1)k_BT}{J}\int\frac{\mathrm{d}^dk}{(2\pi)^d}\frac{1}{a^2 k^2}.

Why: summing the independent modes gives the total angular wandering at a single site.

Step 4. Reduce to a radial integral and analyze the infrared,

\int_{1/L}^{1/a} k^{d-3}\,\mathrm{d}k\ \sim\ \begin{cases}\text{const}&d>2\\\ln(L/a)&d=2\\L^{2-d}&d<2\end{cases}.

Why: the angular measure gives $k^{d-1}$ , which combined with $1/k^2$ is $k^{d-3}$ ; its integral converges at the lower cutoff $1/L\to0$ only when $d>2$ .

Step 5. Insert into the order parameter,

m=\langle\cos\theta\rangle\approx e^{-\frac12\langle\boldsymbol{\pi}^2\rangle}\xrightarrow{d\le2}0.

Why: for $d\le2$ the variance diverges with system size, so the Gaussian suppression drives $m$ to zero — no spontaneous order, exactly Mermin–Wagner. *Verified: verify_aq-sm-mermin-wagner-lower-critical-dimension.py confirms $\int k^{d-3}\mathrm{d}k$ converges iff $d>2$ , gives $\ln(L/a)$ at $d=2$ and $L^{2-d}$ for $d<2$ , and that the resulting $m\to0$ for $d\le2$ .*

Go deeper · Derivation B · 2 minWhy discrete order survives: domain-wall energetics

Derivation BWhy discrete order survives: domain-wall energetics

We want to show, by a Peierls-type counting, why a discrete-symmetry magnet does order for $d\ge2$ — the sharp contrast with the Goldstone case that fixes $d_\ell=1$ (discrete) versus $d_\ell=2$ (continuous). The strategy: estimate the free-energy cost of disordering a discrete magnet by inserting a single domain wall, comparing its energy (proportional to the wall's area, $L^{d-1}$ ) against its entropy of wandering, and see for which dimensions the energy wins as $L\to\infty$ . The key contrast to keep in view is $L^{d-1}$ (a wall) versus $L^{d-2}$ (a smooth twist).

Step 1. A domain wall separating $\pm$ regions costs an energy proportional to its area,

\Delta E\sim \sigma L^{d-1},

with $\sigma\sim J/a^{d-1}$ the surface tension. Why: only bonds crossing the wall are unsatisfied, and their number scales as the $(d-1)$ -dimensional area of the wall.

Step 2. The wall can wander, giving an entropy that grows with its linear extent but more slowly than the energy for $d\ge2$ ,

S_{\text{wall}}\sim k_B\,(d-1)\frac{L}{a}\ln(\text{configs}).

Why: the number of wall shapes grows exponentially in its length, but the energy grows with the full area — a mismatch that favors order in high enough $d$ .

Step 3. Balance energy and entropy,

\Delta F=\Delta E-T\Delta S\sim L^{d-1}\big(\sigma-cT\big).

Why: for $d\ge2$ the leading term $L^{d-1}$ has a positive coefficient at low $T$ , so $\Delta F\to+\infty$ as $L\to\infty$ — walls are suppressed and order is stable.

Step 4. In $d=1$ , $\Delta E$ is a finite constant (a kink is a point) while $\Delta S\sim\ln L$ grows, so $\Delta F=\text{const}-Tk_B\ln L\to-\infty$ for any $T>0$ : order is destroyed, giving $d_\ell=1$ for the discrete case. Compare the continuous case, where a smooth twist costs only $\sim L^{d-2}$ , vanishing at $d=2$ — this softer cost is exactly why $d_\ell$ jumps from $1$ to $2$ when the symmetry becomes continuous. Why: the whole difference between $d_\ell=1$ and $d_\ell=2$ is $L^{d-1}$ (wall) versus $L^{d-2}$ (twist); the extra power of $L$ is what protects discrete order in two dimensions. *Verified: verify_aq-sm-mermin-wagner-lower-critical-dimension.py confirms the wall free energy $\Delta F\sim L^{d-1}(\sigma-cT)$ diverges positively for $d\ge2$ at low $T$ , that the $d=1$ balance $\text{const}-Tk_B\ln L$ goes negative for all $T>0$ (giving $d_\ell=1$ ), and the twist scaling $L^{d-2}$ .*

Limiting cases

How the infrared integral $\int\mathrm{d}^dk/k^2$ behaves controls everything.

Case	Result	Meaning
$d>2$ (e.g. 3D)	$\langle\theta^2\rangle\sim\int_0 k^{d-3}\mathrm{d}k<\infty$	Infrared-convergent: fluctuations bounded, long-range order survives. Goldstone modes merely reduce the order parameter.
$d=2$ (marginal)	$\langle\theta^2\rangle\sim\int\frac{\mathrm{d}k}{k}\sim\ln(L/a)\to\infty$	Logarithmically divergent. True order destroyed, but slowly — correlations decay as a power law, enabling the BKT quasi-ordered phase.
$d=1$	$\langle\theta^2\rangle\sim\int\frac{\mathrm{d}k}{k^2}\sim L\to\infty$	Linearly divergent — no order for continuous or discrete symmetry at $T>0$ (discrete survives only at $T=0$ ).
Long-range coupling $J(r)\sim r^{-(d+\sigma)}$	$\epsilon(k)\sim k^\sigma\ (\sigma<2)$	A stiffer infrared dispersion can render the integral convergent in $d=2$ , restoring order — the theorem assumes short-range interactions.

Worked examples

Example 1 — How much bigger a 2D system must be to double its angular fluctuation.

Step 1. In two dimensions the angular variance grows as $\langle\delta\theta^2\rangle=\frac{k_BT}{2\pi J}\ln(L/a)$ — a logarithm of system size.

Step 2. Take $L_1=100\,a$ and $L_2=10^4\,a$ . The ratio of variances is $\ln(L_2/a)/\ln(L_1/a)=\ln(10^4)/\ln(10^2)=4\ln10/(2\ln10)=2$ .

Step 3. So doubling the fluctuation requires squaring the linear system size (from $100a$ to $10{,}000a$ ). The divergence is real but painfully slow — which is exactly why 2D quasi-order looks ordered on any finite sample.

Step 4. Contrast 3D, where $\langle\delta\theta^2\rangle$ saturates: increasing $L$ from $100a$ to $10^4a$ changes it by a negligible amount, and order is robust. To double the 2D angular fluctuation you must square the system size ( $100a\to10{,}000a$ ), because $\langle\delta\theta^2\rangle\propto\ln L$ ; in 3D it barely changes.

Example 2 — Reading the lower critical dimension off the infrared integral.

Step 1. The convergence of $\int_{1/L} k^{d-3}\mathrm{d}k$ at $k\to0$ decides order. The primitive is $k^{d-2}/(d-2)$ for $d\neq2$ and $\ln k$ for $d=2$ .

Step 2. For $d=3$ : $\int_{1/L}k^0\mathrm{d}k=[k]_{1/L}$ , finite as $L\to\infty$ — order survives.

Step 3. For $d=2$ : $\int_{1/L}k^{-1}\mathrm{d}k=[\ln k]_{1/L}=\ln L$ , diverges (marginally) — no continuous order, quasi-order possible.

Step 4. For $d=1$ : $\int_{1/L}k^{-2}\mathrm{d}k=[-k^{-1}]_{1/L}\sim L$ , diverges linearly — strongly disordered. Hence continuous symmetry has $d_\ell=2$ : the integral first diverges at $d=2$ . The lower critical dimension for continuous symmetry is $d_\ell=2$ , because $\int k^{d-3}\mathrm{d}k$ first diverges (logarithmically) exactly at $d=2$ .

Origins · 1 minMermin-Wagner theorem and how this was discovered

Mermin-Wagner theorem

proved 1966 · The rigorous statement that continuous symmetries cannot be spontaneously broken in dimensions d ≤ 2 at nonzero temperature with short-range interactions, proved by N. David Mermin and Herbert Wagner in 1966 using the Bogoliubov inequality. Pierre Hohenberg established the companion result for superfluids and superconductors in 1967, and Sidney Coleman gave a field-theory version in 1+1 dimensions in 1973. The theorem is a cornerstone of the modern understanding of ordering, dimensionality, and the special marginality of two dimensions.

History

1958

— Bogoliubov's inequality provides the operator bound later used in the proof

1966

— Mermin and Wagner prove the theorem for Heisenberg magnets using the Bogoliubov inequality

1967

— Hohenberg extends the result to superfluids and superconductors (no 2D condensate at T > 0)

1973

— Coleman gives a field-theory version forbidding Goldstone bosons in 1+1 dimensions

1973

— Kosterlitz and Thouless show the 2D XY model still has a transition despite the theorem

Going further

References. Mermin & Wagner, Phys. Rev. Lett. 17, 1133 (1966) · Hohenberg, Phys. Rev. 158, 383 (1967) · Kardar, Statistical Physics of Fields, Ch. 3 · Pathria & Beale, Statistical Mechanics, 3rd ed. · Chaikin & Lubensky, Principles of Condensed Matter Physics, Ch. 6 · Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity.

Provenance. The harmonic spin-wave estimate and the $\int\mathrm{d}^dk/k^2$ infrared classification follow Kardar (Statistical Physics of Fields, Ch. 3) and Chaikin & Lubensky (Ch. 6); verify_aq-sm-mermin-wagner-lower-critical-dimension.py recomputes that $\int k^{d-3}\mathrm{d}k$ converges iff $d>2$ , with $\ln L$ at $d=2$ and $L^{2-d}$ for $d<2$ . The rigorous Bogoliubov-inequality bound is from Mermin & Wagner (1966) and Hohenberg (1967). The domain-wall energetics ( $\Delta E\sim L^{d-1}$ ) and $d_\ell=1$ for the discrete case follow Peierls' argument and the transfer-matrix solution of the 1D Ising chain (Pathria & Beale). The 2D $XY$ exponent $\eta(T_{\mathrm{BKT}})=1/4$ and the Nelson–Kosterlitz universal jump are standard BKT results (Kosterlitz–Thouless 1973; Nelson–Kosterlitz 1977) and are used here by name.

Builds on: Order Parameter and Spontaneous Magnetization · The Ising Model: Definition and Hamiltonian · Peierls Argument for Long-Range Order · One-Dimensional Ising Model and Transfer Matrix Method. Leads to: Domain Walls and Interfacial Excitations · High- and Low-Temperature Series Expansions.

If spin waves cost so little energy, why does 3D order at all?

It is not the cost of one spin wave but the sum over all of them that matters. The angular variance is $\langle\theta^2\rangle\sim\int\mathrm{d}^dk/k^2\sim\int k^{d-3}\mathrm{d}k$ . In three dimensions this integral converges at small $k$ , so the accumulated fluctuation is finite and the magnetization is merely reduced, not destroyed. In two dimensions it grows as $\ln L$ , and in one as $L$ , so the fluctuations pile up and average the order to zero. Cheap modes are harmless when there are few of them (high $d$ ) and fatal when there are many (low $d$ ) — a matter of phase space.

Is the harmonic (spin-wave) argument actually a proof, or just an estimate?

The spin-wave calculation is a heuristic estimate — it assumes small fluctuations and a Gaussian theory. The proof is Mermin and Wagner's, and it uses the Bogoliubov inequality to bound the magnetization by an integral of the form $\int\mathrm{d}^dk/(h+k^2)$ with no harmonic approximation, valid for the full interacting model. Reassuringly, that rigorous integral has exactly the same infrared structure as the spin-wave estimate: it diverges as the field $h\to0$ for $d\le2$ , forcing $m\to0$ . So the intuitive picture and the theorem agree, but only the Bogoliubov bound is airtight.

Does Mermin–Wagner apply at zero temperature?

No — the theorem is explicitly a finite-temperature statement; it relies on thermal occupation of the Goldstone modes, which vanishes at $T=0$ . At absolute zero a 2D Heisenberg antiferromagnet can and does have long-range order (quantum order), and 2D quantum systems routinely order in their ground states. The theorem forbids thermal breaking of continuous symmetry for $T>0$ in $d\le2$ ; the quantum ground state is governed by different (quantum, $d+1$ -dimensional) considerations.

What is special about d = 2 rather than d = 3 or d = 1?

Two dimensions is the lower critical dimension for continuous symmetry — the exact borderline where the infrared integral $\int k^{d-3}\mathrm{d}k$ switches from convergent to divergent. For $d>2$ it converges (order), for $d<2$ it diverges as a power (strong disorder), and at $d=2$ it diverges only logarithmically. That marginal logarithm is why 2D is not simply disordered: correlations decay as slow power laws (quasi-long-range order) rather than exponentially, opening the door to the topological BKT transition. Two dimensions is delicately poised between order and disorder.

Why does a domain wall save discrete order but a spin wave does not?

A domain wall in a discrete magnet is a genuine boundary whose energy scales with its area, $\sigma L^{d-1}$ — for $d\ge2$ this grows with system size, so making one becomes prohibitively costly and order is stable. A continuous magnet has no such wall; its cheapest large-scale excitation is a smooth twist costing only $\sim L^{d-2}$ , which vanishes at $d=2$ . The extra power of $L$ (area $L^{d-1}$ versus twist $L^{d-2}$ ) is exactly the difference between $d_\ell=1$ and $d_\ell=2$ . Discreteness gives you a rigid, expensive wall; continuity gives you a soft, cheap twist.

How do 2D superconductors and superfluids exist if there is no condensate?

Hohenberg's theorem forbids a true condensate (broken $U(1)$ phase symmetry) in 2D at $T>0$ , just as Mermin–Wagner forbids 2D magnetic order. But superfluidity and superconductivity do not require true long-range order — they require only quasi-long-range order with finite phase stiffness, which the 2D system has below $T_{\mathrm{BKT}}$ . The phase correlations decay as a slow power law, the superfluid density is finite, and it jumps by the universal Nelson–Kosterlitz value at the BKT transition. So you get dissipationless flow without a genuine condensate — a subtlety unique to two dimensions.